--- title: "BL growth along thin plate, examples" subject: "Fluid Mechanics" chapter: 8 key: "semester_3/fluid_mechanics:bl-growth-thin-plate-examples" source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/bl-growth-thin-plate-examples site: BCT Engineering --- # BL growth along thin plate, examples Fluid Mechanics · Chapter 8: Boundary Layer theory - Asked in: 2072 Chaitra, 1+3+1 marks - Video: [Boundary Layer Theory - Introduction](https://www.youtube.com/watch?v=54zf68IoPWU) — TutorialsPoint - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/bl-growth-thin-plate-examples - Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-8 ## Question Define the concept of boundary layer. Explain the growth of boundary layer along a thin plate, when liquid is flowing over it, both for laminar and turbulent flow. Give two examples of use of boundary layer concept. ## Short answer The boundary layer is the thin viscous region next to a solid surface where velocity changes from zero to the free-stream value. On a thin plate, laminar flow near the leading edge gives δ proportional to sqrt(x); after transition (Re_x ≈ 5×10^5) turbulent flow makes δ grow faster, proportional to x^0.8. Applications include skin-friction drag calculation for ships and wings, and designing streamlined shapes to delay separation. ## Answer #### Concept of boundary layer The boundary layer is the thin region next to a solid surface where the velocity changes from zero at the surface (no-slip) to the free-stream value $U$, because of viscosity. Outside it the flow is effectively inviscid. #### Growth along a thin plate ```text U -> ------------------------------------------ laminar transition turbulent ------------___________.........______________________ delta ~ sqrt(x) delta ~ x^0.8 ========================================= plate --> leading edge ``` **Laminar flow (near the leading edge, $Re_x < 5\times10^5$)** - The layer starts with zero thickness at the leading edge and grows slowly as viscous effects spread outward. - The fluid moves in smooth layers. Blasius's result: $\delta = \dfrac{5x}{\sqrt{Re_x}}$, so $\delta \propto \sqrt{x}$. - The velocity profile is nearly parabolic, with a gentle gradient at the wall. **Turbulent flow (downstream, $Re_x > 5\times10^5$)** - After a transition region the flow becomes turbulent and eddies mix the fluid strongly. - The layer is thicker and grows faster: $\delta = \dfrac{0.37x}{Re_x^{1/5}}$, $\delta \propto x^{0.8}$. - The profile is fuller ($u/U = (y/\delta)^{1/7}$), the wall shear stress is larger, and a thin laminar sub-layer stays at the wall. Here $Re_x = Ux/\nu$. #### Two examples of use 1. Finding the **skin-friction drag** on ship hulls, aircraft wings and flat surfaces. 2. Designing **streamlined shapes** to delay separation and reduce drag, and wind tunnel sections using displacement thickness. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How does laminar boundary layer thickness vary with x? A. delta proportional to x^0.8 B. delta constant C. delta proportional to x^2 D. delta = 5x/sqrt(Re_x), proportional to sqrt(x) Answer: D. This is Blasius's result. (Key point: Laminar growth) 2. How does turbulent thickness vary? A. delta proportional to sqrt(x) B. delta proportional to 1/x C. delta = 0.37x/Re_x^0.2, proportional to x^0.8 D. delta is independent of x Answer: C. The turbulent layer thickens faster than the laminar. (Key point: Turbulent growth) 3. What is the turbulent velocity profile? A. Parabolic with lower wall shear B. Fuller, approximated by u/U = (y/delta)^(1/7), with higher wall shear C. Linear from wall to edge D. Uniform velocity Answer: B. Eddy mixing flattens the profile and increases wall gradient. (Key point: Turbulent profile) 4. What decides when the layer is laminar or turbulent? A. Re_x compared with about 5 x 10^5 B. Plate temperature C. Plate colour D. Fluid depth Answer: A. Laminar for Re_x < 5 x 10^5 and turbulent beyond. 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